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Caveman Mathematics And The Primordial Earth!
By!
Ian Beardsley!
Copyright © 2021 by Ian Beardsley"
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One of the mysteries of how life on earth began is how the amino acids, the building blocks of
life were made. Miller and Urey showed if they mixed together the substances of the primordial
earth methane (CH4) ammonia (NH3) and water (H2O) under electricity (to mimic electrical
storms) they could produce eleven of the twenty amino acids present in proteins.!
What better way to understand the primordial earth than by way of the primordial caveman?!
Thus at first glance how does a caveman look at the structure of the primordial gas methane
(CH4)? Well CH4 is carbon (C) which is (has 4 valence electrons for bonding). And, is H4
which means there are 4 hydrogen atoms that connect to it. But hydrogen (H) is meaning
for CH4 to be neutral it needs . So we have the chemical structure of methane is:!
!
A carbon surrounded by by four hydrogens in single bonds. The
caveman asks how four things fit around one? Being smart he says
a square tiled by four regular octagons:!
!
Never mind that a carbon atom C is much larger than a hydrogen
atom H…!
So therefore it would be more accurately represented by a carbon atom at the center
surrounded four hydrogen atoms as squares, the carbon atom the regular octagon, the exact
inversion:!
Which actually closely represents the relative sizes of carbon and
hydrogen if they are circles inscribed in them.!
Let us compute the scenario of our caveman’s desire to see carbon tiled by hydrogen without
leaving gaps:"
C
4−
H
1+
4H
+
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!
For the square the hydrogen has a radius given by the
apothem, a, of the square. The diagonal of the square is
so the radius of it is .!
!
!
!
It is obvious from the drawing that the apothem is 1/2, but we are warming-up to calculating
the apothem for a regular octagon. It is…!
!
The apothem of a regular octagon is the diameter d divided
by 2. The side of the octagon is 1. Because it connects with
the square. We have if the side is s, that…!
!
!
!
!
!
Thus…!
r
H
2
2/2
(
1
2
)
2
+ a
2
=
(
2
2
)
2
1
4
+ a
2
=
2
4
a =
1
2
d = s + 2x
x
2
+ x
2
= s
2
2x
2
= 1
2
x
2
=
1
2
x =
2
2
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!
!
!
!
Is the apothem of the regular octagon.!
We then ask, how can methane be a central carbon as a square surrounded by four regular
octagons and be a regular octagon surrounded four hydrogens. One is much larger than the
other? Let us say they are oscillating between two states as the caveman would have it:!
1 + 2x = d
1 + 2
2
2
= d
d = 1 + 2
a =
2 + 1
2
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So if the apothem (a) of the regular octagon becomes the apothem of the square then the
regular octagon expands greatly, as seen here…!
!
Thus if…!
!
!
!
!
"
a =
2 + 1
2
⟶ a
2
+ a
2
=
(
s
2
)
2
( 2 + 1)
2
=
s
2
4
4(2 2 + 3) = s
2
s = 8 2 + 12
s = 4.83
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The the apothem of the regular octagon becomes…!
The apothem has gone from 1.207 to 4.1221 and this is an increase in size of…"
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!
!
!
!
!
!
!
!
!
!
But, the increase in size does not have to be by the apothem of the regular octagon becomes
the apothem of the square, but rather this by a factor of the inverse of the golden ratio. Our
caveman asks if I divide a line such that…!
d = 1 + 2
2d = 2(1 + 2)
d = 2 + 2 2
2d = 5.414
d = 8.2442
2a = 2
( 2 + 1)
2
2 ⋅ 2a = 2( 2 + 1)
4a = 2 2 + 2
4a + d = 4.828 + 8.2442 = 13.0726
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!
where !
Then,…!
!
!
!
!
!
!
The golden ratio conjugate.!
So we have,…!
!
Times larger. We have considered the primordial earth gas CH4 (methane) now let us go to
NH3 (ammonia). The caveman suggests the oscillations in the state from CH4 as regular
octagons surrounding a square to regular octagons surrounded by squares is really the change
from air, which is N2 and O2 to the primordial gas NH3. Dry air at standard temperature and
pressure at sea level 22 degrees C and 1.013 bar is a density of 1.225 kilograms per cubic
meter. Ammonia gas (NH3) at boiling point and 1.013 bar is 0.86 kilograms per cubic meter.
Thus,…!
!
Is approximately our 1.5 above. This is more exactly an accuracy of .
Which is 95.456%"
a
b
=
b
c
a = b + c
a c = b
2
c = a − b
a(a − b) = b
2
a
2
b
2
−
a
b
− 1 = 0
a
b
= Φ =
5 + 1
2
ϕ =
1
Φ
=
b
a
=
5 − 1
2
13.0726
5.414
ϕ = 2.41(0.618) = 1.5
1.225
0.86
= 1.4
1.4244
1.4922
= 0.95456
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Thus, what has our caveman done?!
!
He has shown that dry air to NH3 (ammonia) is the geometric inverse by density of methane
(CH4).!
He has found the geometric composition of the primordial earth.!
We have considered the primordial earth CH4 and NH4, and that leaves H2O (water). This is
simple. With the advent of photosynthesizers the earth filled up with oxygen (they converted
carbon dioxide (CO2) to oxygen gas O2). The present earth atmosphere is basically about 25%
O2 and 75% N2. O=16.00 g/mol and N=14.01 g/mol. Thus air as a mixture is:!
0.25O2 + 0.75N2=air=29.0 g/mol!
H2O=2.02+16.00=18.02 g/mol!
Thus,…!
!
The Golden Ratio."
air
H
2
O
=
29.0
18.02
= 1.61 ≈ Φ
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The Author!
